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Topic: It is shown that ZFC is inconsistent
Replies: 0

 byron Posts: 865 Registered: 3/3/09
It is shown that ZFC is inconsistent
Posted: Feb 6, 2013 6:46 AM

It is shown that ZFC is inconsistent

in ZFC IS AN AXIOM CALLED THE Axiom schema of specification

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

3. Axiom schema of specification (also called the axiom schema of
separation or of restricted comprehension): If z is a set, and \phi\! is
any property which may characterize the elements x of z, then there is a
subset y of z containing those x in z which satisfy the property. The
"restriction" to z is necessary to avoid Russell's paradox and its
variant

the axiom of separation is used to outlaw/block/ban impredicative statements like Russells paradox

now
the axiom of separation of ZFC is it self impredicative as Solomon Ferferman points out

http://math.stanford.edu/~feferman/papers/predicativity.pdf

"in ZF the fundamental source of impredicativity is the seperation axiom
which asserts that for each well formed function p(x)of the language ZF
the existence of the set x : x } a ^ p(x) for any set a Since the formular
p may contain quantifiers ranging over the supposed "totality" of all the
sets this is impredicativity according to the VCP this impredicativity is given teeth by the axiom of infinity

thus it outlaws/blocks/bans itself
thus ZFC contradicts itself and 1)ZFC is inconsistent 2) that the paradoxes it was meant to avoid are now still valid and thus mathematics is inconsistent